 Solution

For a rough boundary, integrate the similarity shear from the roughness length. A usual moderate-stability closure iswith positive dimensionless coefficients. These are empirical closures, not quantities determined by similarity alone. An often-used choice is $c_s=5,c_u=16$. The unstable quarter-power relation is tested over approximately $-1\lesssim z/{\ell }_O\lesssim -0.01$ in Dyer's flux-gradient study; the stable coefficient convention is recorded in the surface-flux formulation. Near neutrality one takes their smooth limits.

For moderate stable conditions, integration givesFor the unstable closure set $x(\zeta )=(1-c_u\zeta {)}^{1/4}$ and defineThen $1-\zeta {\psi }_m^{\prime }(\zeta )={\varphi }_m(\zeta )$ andFor weak unstable departure, $|c_{u}z/{\ell }_O|\ll 1$, this reduces to the stable-form expression with $c_s$ replaced by $c_u/4$ and a negative ${\ell }_O$. The full quarter-power expression is useful for moderately unstable conditions with $|z/{\ell }_O|$ of order one; it must not be extrapolated automatically to the distinct asymptotic free-convection regime. The stable linear expression similarly describes a continuously turbulent surface layer, rather than guaranteeing validity after strong stratification destroys the constant-flux state. Both profiles require $z_0\ll z\ll {\delta }_{\tau }$ and approximately uniform fluxes.